Ergodic theory of low-dimensional dynamical systems

低维动力系统的遍历理论

• 批准号: RGPIN-2017-06521
• 负责人: Tiozzo, Giulio
• 金额: $ 3.64万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749632
• 关键词: Ergodic; theory; low; dimensional; dynamical; Ergodic; theory; low; dimensional; dynamical

The goal of this project is to investigate dynamical properties of low-dimensional systems using methods from ergodic theory. In particular, I will develop the theory of core entropy of polynomials as recently introduced by W. Thurston. Properties of this function will be related to the geometry and topology of the Mandelbrot set, and will be used to better understand the geometry of the parameter space of polynomials of higher degree and rational maps. Moreover, I will investigate the ergodic theory of random walks on groups acting on spaces with hyperbolic properties, and derive applications to geometric group theory, Teichmueller dynamics and low-dimensional topology. This will extend the theory of random walks on Lie groups, as developed by Furstenberg, Margulis, Zimmer, and others, to groups of interest in geometry and topology such as the mapping class group and the group of outer automorphisms of the free group.

该项目的目的是使用来自厄运理论的方法研究低维系统的动力学特性。特别是，我将根据W. Thurston最近引入的多项式核心熵理论。该功能的属性将与Mandelbrot集的几何形状和拓扑相关，并将用于更好地了解高度和理性图的多项式参数空间的几何形状。此外，我将研究作用于具有双曲线特性空间的群体上的随机步行理论，并将应用于几何组理论，Teichmueller动力学和低维拓扑。这将把随机步行的理论扩展到弗斯滕伯格，马古利斯，齐默尔等人开发的谎言群体上，到了几何学和拓扑的群体，例如映射类群和自由组的外部自动形态。